HP-discontinuous galerkin method for nonlinear problems

hp-DISCONTINUOUS GALERKIN METHOD FOR NONLINEAR PROBLEMS V´ ı Dolejˇ s´ ı Cha les Uni e si y P ague Facul y o Ma hema ics and Physics Depa men o Nume ical Ma hema ics Sokolo sk´ a 83, 186 75 P aha, Czech Republic [email p o ec ed] .cuni.cz Abs ac We deal wi h a nume ical solu ion o nonlinea con ec ion-di usion p oblems wi h he aid o he discon inuous Gale kin ini e elemen (DGFE) me hod. We p opose a new hp-adap a ion echnique, which is based on a combina ion o a esiduum-noncon o mi y es ima o and a e- gula i y indica o . The esiduum-noncon o mi y es ima o consis s o wo building blocks ( he esiduum e o indica o and he alue o he noncon o mi y). The es ima o ma ks mesh ele- men s o a e inemen . The egula i y indica o decides i he ma ked elemen s will be e ined by h- o p- echnique. The esiduum-noncon o mi y es ima o as well as he egula i y indica o a e easily compu able quan i ies. Mo eo e , he same echnique es ima es an algeb aic e o a ising om an i e a i e solu ion o he co esponding nonlinea algeb aic sys em. The pe o - mance o he p oposed hp-DGFE me hod is demons a ed by se e al nume ical examples. Keywo ds: hp-discon inuous Gale kin ini e elemen me hod; esiduum-noncon o mi y indi- ca o ; egula i y es ima o ; algeb aic e o . In oduc ion Ou aim is o de elop a su icien ly obus , e icien and accu a e nume ical scheme o he simula ion o iscous comp essible lows. The discon inuous Gale kin ini e elemen (DGFE) me hods ha e become e y popula nume ical echniques o he solu ion o he comp essible Na ie -S okes equa ions. Recen p og ess o he use o he DG me hod o comp essible low simula ions can be ound in [11]. In his pape , we deal wi h a model p oblem ep esen ed by a scala nonlinea con ec ion- di usion equa ion. We ocus on a hp-adap a ion s a egy o DGFE me hods which signi ican ly inc ease he accu acy and e iciency o he compu a ion, see, e.g. [2, 6, 8, 12, 13]. The p oposed s a egy is based on a combina ion o a esiduum-noncon o mi y es ima o and a egula i y indica o . The esiduum-noncon o mi y es ima o gi es a lowe es ima e o he e o measu e consis ing o he e o measu ed in a dual no m and he quan i y measu ing a iola ion o he con o mi y o he solu ion. This es ima o is locally de ined o each mesh elemen , i is easily compu able and i s implemen a ion is e y simple. The egula i y indica o is based on he in eg a ion o in e elemen jumps o he app oxima e solu ion o e he elemen bounda y. Taking in o accoun esul s om a p io i e o analysis, we de ine he egula i y indica o . I his alue is smalle han one hen we apply a p- e inemen o he wise we use ah- e inemen . Bo h e inemen s (hand p) a e only iso opic, an aniso opic adap a ion will be a subjec o he u he esea ch. 56 1 P oblem Fo mula ion 1.1 Go e ning Equa ions We conside a o mal nonlinea pa ial di e en ial equa ion Lu =0 in Ω,(1a) u=uDon ∂ ΩD,(1b) Nu =gNon ∂ ΩN,(1c) whe e u:Ω→Ris he unknown scala unc ion de ined on Ω∈Rd,d=2,3, Lis a o mal second o de di e en ial ope a o and (1b) and (1c) o mally ep esen he Di ichle and Neu- mann bounda y condi ions on he pa s o bounda y ∂ ΩDand ∂ ΩN, espec i ely. We assume ha he e exi s a unique weak solu ion o (1) which we deno e again by u. Le Th(h>0) be a pa i ion o he closu e Ωo he domain Ωin o a ini e numbe o closed d-dimensional simplicies Kwi h mu ually disjoin in e io s. We call Th={K}K∈Tha iangu- la ion o Ωand do no equi e he con o ming p ope ies om he ini e elemen me hod. O e he iangula ion Thwe de ine he so-called b oken Sobole space Hs(Ω,Th):={ ; |K∈Hs(K)∀K∈Th},s≥0 (2) wi h he semino m | |Hs(Ω,Th):=∑K∈Th| |2 Hs(K)1/2,whe e |·|Hs(K)deno es he semino m o he Sobole space Hs(K),K∈Th. Mo eo e , o each K∈Th, we assign a posi i e in ege pK(=local polynomial deg ee). Then we de ine he se ph:={pK,K∈Th}.(3) We de ine he ini e dimensional subspace o H1(Ω,Th)which consis s o discon inuous piece- wise polynomial unc ions associa ed wi h he ec o phby Shp={ ; ∈L2(Ω), |K∈PpK(K)∀K∈Th},(4) whe e PpK(K)deno es he space o all polynomials on Ko deg ee ≤pK,K∈Th. We in oduce a o mal disc e iza ion o (1) wi h he aid o DGFE me hod. Hence, le ˜ch(u, ):H2(Ω,Th)×H2(Ω,Th)→R(5) be he co esponding o m which is nonlinea wi h espec o i s i s a gumen and linea wi h espec o he second one. We say ha he unc ion uh∈Shpis an app oxima e solu ion o (1), i ˜ch(uh, h) = 0∀ h∈Shp.(6) Mo eo e , we de ine he o m Nh:H1(Ω,Th)→Rby Nh( ):= 2∑ Γ∈FI hZΓh−1 Γ[[ ]]2dS+∑ Γ∈FD hZΓh−1 Γ( −uD)2dS  1/2 ,(7) whe e uDis om (1b), FI hdeno es he se o all in e io aces o Th,FD hdeno es he se o all aces o Thlying on ∂ ΩD,[[·]] is he jump o a unc ion om H1(Ω,Th)and hΓis he diame e o a ace Γ. Finally, we cha ac e ise he weak solu ion o (1). 57 Lemma 1.1 The ollowing implica ions a e alid: i) Le u ∈H2(Ω)be he weak solu ion o (1) hen ˜ch(u, ) = 0∀ ∈H2(Ω,Th),(8a) Nh(u) = 0.(8b) ii) I u ∈H2(Ω,Th)sa is ies bo h condi ions o (8) hen u is he weak solu ion o (1). The disc e e p oblem (6) ep esen s a sys em o Nh=dimShpnonlinea algeb aic equa ions. We sol e i wi h he aid o a New on-like i e a i e me hod which gi es he solu ion ˜uh∈Shp such ha ˜ch(˜uh, h)≈0∀ h∈Shp. 2 Residuum Es ima o In his sec ion we in es iga e he disc e iza ion e o u−uhand he algeb aic e o ˜uh−uhin a sui able (dual) no m and de ine es ima o s gi ing some in o ma ion abou hese e o s. 2.1 E o Measu e Simila ly as in [3], ou p oposed e o measu e consis s o wo building blocks, which a e mo- i a ed by Lemma 1.1, namely ela ions (8a) and (8b). Le X:=H2(Ω,Th)and k·kXis a no m de ined on X, which will be speci ied la e . The i s building block is gi en by Rh(uh):=sup 06= ∈X ˜ch(uh, ) k kX(9) which de ines he esiduum e o in he dual no m o he app oxima e solu ion uh∈Shp⊂X and i measu es a iola ion o (8a). Howe e , i is impossible o e alua e Rh(uh), since he sup emum is aken o e an in ini e-dimensional space. The e o e, in ou app oach, we seek he maximum o e some su icien ly la ge bu ini e dimension subspace o X, which is p esen ed in Sec ion 2.2. The second building block is based on (8b), which cha ac e ises a iola ion o he con o mi y o he weak solu ion and a iola ion o he Di ichle bounda y condi ion. I is ep esen ed by he alue Nh(uh)≥0 gi en by (7) which we call he noncon o mi y o he app oxima e solu ion. In con as o Rh(uh) he quan i y Nh(uh)is di ec ly compu able om (7). Finally, ou e o measu e is he sum o squa es o he esiduum e o and noncon o mi y, i.e., Eh(uh):= (Rh(uh)2+Nh(uh)2)1/2.(10) Due o Lemma 1.1 we simply obse e ha Eh(uh) = 0 i and only i uh=u. 2.2 Global and Elemen Residuum Es ima o s In he p e ious sec ion, we in oduced he e o measu e Eh(uh) = pRh(uh)2+Nh(uh)2o he app oxima e solu ion uh∈Shp⊂X. Whe eas Nh(uh)is easy o e alua e, he quan i y Rh(uh) has o be app oxima ed in a sui able way, which is p esen ed in his sec ion. Fo each K∈Th and each in ege p≥0, we de ine he spaces Sp K:={ φ h∈X, φ h|K∈Pp(K), φ h|Ω K=0}(11) 58 and S+ hp:={ φ ∈X; φ =∑ K∈Th cK φ K,cK∈R, φ K∈SpK+1 K,K∈Th}.(12) Ob iously, Shp⊂S+ hp⊂X. Now, we de ine he elemen esiduum es ima o ρ h,K(uh):=sup 06= ψ h∈SpK+1 K ˜ch(uh, ψ h) k ψ hkX=sup ψ h∈SpK+1 K,k ψ hkX=1 ˜ch(uh, ψ h),uh∈X,(13) o each K∈Thand he global esiduum es ima o ρ h(uh):=sup 06= ψ h∈S+ hp ˜ch(uh, ψ h) k ψ hkX=sup ψ h∈S+ hp,k ψ hkX=1˜ch(uh, ψ h),uh∈X,(14) which a e easily compu able quan i ies i k·kXis sui ably chosen, see Sec ion 2.4. Ob iously, i u∈Xis he exac solu ion o (1) hen consis ency (8a) implies 0 = ρ h(u) = ρ h,K(u),K∈Th. Mo eo e , we ha e immedia ely a lowe bound ρ h(uh)≤Rh(uh),(15) since ρ his he sup emum o e subspace S+ hp⊂X. Howe e , i is open i he e exis s an uppe bound, i.e., Rh(uh)≤C ρ h(uh), whe e C>0. This will be he subjec o a u he esea ch. 2.3 Algeb aic Residuum Es ima o s Simila ly as in he p e ious sec ion, we de ine he es ima o co esponding o he algeb aic e o esiduum. Le ˜uh∈Shpbe he ou pu o he i e a i e p ocess used o he solu ion o (6). We de ine he algeb aic esiduum es ima o ρ A h(˜uh):=sup 06= ψ h∈Shp ˜ch(˜uh, ψ h) k ψ hkX=sup ψ h∈Shp,k ψ hkX=1˜ch(uh, ψ h),(16) which measu es he algeb aic e o ( he di e ence be ween ˜uhand uh), since ρ A h(uh) = 0 due o (6). The ela ions (14) and (16) gi e ρ A h(˜uh)≤ ρ h(˜uh), since in (14) he sup emum is aken o e he la ge space. We s op he i e a i e p ocess o he solu ion o he nonlinea algeb aic p oblem (6) i ρ A h(˜uh)≤ βρ h(˜uh),(17) whe e β ∈(0,1). In nume ical expe imen s we pu β ≈0.01. 2.4 Choice o he No m k·kX In o de o ensu e a as e alua ion o es ima o s ρ hand ρ A h, we need o choose he no m k·kX in a sui able way. We employ k·kX:= δ k·k2 L2(Ω)+ ε |·|2 H1(Ω,Th)1/2,(18) which gua an ees ha ρ h(uh)2=∑ K∈Th ρ h,K(uh)2.(19) He e δ and ε deno e he size o “con ec ion” and “di usion”, espec i ely. The e o e, i is su icien o e alua e he elemen esiduum es ima o s ρ h,K,K∈Th. This is a s anda d ask o seeking a cons ain ex ema o e SpK+1 Kwi h he cons ain k ψ hkX=1. This can be done di ec ly e y as since he dimension o SpK+1 K,K∈This small. 59 2.5 Residuum-Noncon o mi y Es ima o s We ha e al eady men ioned ha he second building block o he e o measu e is gi en by he noncon o mi y Nh(uh)de ined by (7). Fo he pu pose o a mesh adap a ion, we de ine i s local a ian Nh,K( ):= ∑ Γ∈FI h∩ ∂ KZΓh−1 Γ[[ ]]2dS+∑ Γ∈FD h∩ ∂ KZΓh−1 Γ( −uD)2dS  1/2 , ∈H1(Ω,Th). (20) Ob iously, om (7) and (20), we ha e Nh( )2=∑K∈ThNh,K( )2.Finally, we de ine he local and global esiduum-noncon o mi y es ima o s o he app oxima e solu ion uh∈Shpby η h,K(uh):= ρ h,K(uh)2+Nh,K(uh)21/2,K∈Th,(21a) and η h(uh):= ρ h(uh)2+Nh(uh)21/2= ∑ K∈Th η h,K(uh)2!1/2 ,(21b) espec i ely. In i ue o (10), (15) and (21), we expec ha he global esiduum-noncon o mi y es ima o η h(uh)app oxima es he e o measu e Eh(uh). In he ollowing we in oduce an adap- a ion echnique which p oduces a hp-mesh and he co esponding app oxima e solu ion such ha he es ima o η h(uh)is unde a gi en ole ance. 3hp-Adap a ion P ocess In his sec ion, we p esen a new hp-adap i e DG echnique o he solu ion o (6). In Sec ion 2, we de ined he elemen and global esiduum-noncon o mi y es ima o s η h,Kand η h, espec- i ely. We employ he no m k·kXgi en by (18) which gua an ees ha equali y (19) is alid. As al eady men ioned, ou in e es is o ind he solu ion ˜uh∈Shpsuch ha η h(˜uh)≤ ω ,(22) whe e ω >0 is a gi en ole ance. Le Thbe a gi en mesh and ˜uh he co esponding app oxima ion o (6). We equi e ha η h,K(˜uh)≤ ω √#Th∀K∈Th,(23) whe e #Thdeno es he numbe o elemen s o Th. Ob iously, i (23) is sa is ied hen, due o (21b), condi ion (22) is alid and he adap a ion p ocess s ops. O he wise, we ma k o e inemen all K∈Th iola ing (23). Fu he mo e, all ma ked elemen s will be e ined ei he by h- o by p-adap a ion, namely, ei he we spli a gi en mo he elemen Kin o ou daugh e elemen s o we inc ease he deg ee o polynomial app oxima ion o a gi en elemen . Thus he new mesh Tˆ hand new se {ˆpK,K∈ Tˆ h}a e c ea ed. We in e pola e he old solu ion on a new mesh and pe o m he nex adap a ion s ep ill (22) is alid. 3.1 Regula i y Indica o The es ima ion o he egula i y o he solu ion is an essen ial key o any hp-adap a ion s a egy. Ou app oach is based on a measu e o in e -elemen jumps which is he base o he jump indica o om [5] and he shock cap u ing echnique om [7]. 60 We p opose he egula i y indica o gK(uh):=R ∂ K∩Ω[[uh]]2dS |K|h2pK−2 K ,K∈Th,(24) whe e |K|is he a ea o K∈Th. I he exac solu ion is su icien ly egula , i.e., sK≥pK+1, hen gK(uh)≈Oh2pK+1 K/h2 Kh2pK−2 K=O(h1 K).On he o he hand, i he exac solu ion is no su icien ly egula , i.e., sK<pK+1 (⇔sK≤pK), hen gK(uh)≈Oh2sK−1 K/h2 Kh2pK−2 K= O(h2 δ −1 K),whe e δ =sK−pk≤0. Then we use he ollowing s a egy gK(uh)≤1⇒solu ion is egula ⇒p- e inemen , gK(uh)>1⇒solu ion is i egula ⇒h- e inemen ,K∈Th.(25) 4 Nume ical Expe imen s In he p e ious sec ions, we in oduced and de eloped he adap i e hp-DGFE me hod. We demons a e i s pe o mance in his sec ion by se e al nume ical examples. Le ˜uhbe he ap- p oxima e solu ion esul ing om an i e a i e me hod, i.e., he solu ion in luenced by he alge- b aic e o . We deal wi h wo ollowing nume ical examples: (E1) nonlinea con ec ion-di usion equa ion wi h a co ne singula i y om [9], (E2) linea con ec ion-di usion equa ion wi h he s ong in e io laye and he exponen ial bounda y laye om [10]. Fo he i s one, we know he exac solu ion and he e o e we a e able o e alua e he compu a ional e o . We ca ied ou a hp-adap i e algo i hm s a ing on a mesh wi h he s ep h0and P1polynomial app oxima ion. We e alua e ku−˜uhkX,Nh(˜uh)and ρ h(˜uh)wi h he co esponding expe imen al o de s o con e gence (EOC) wi h espec o he numbe o deg ee o eedom Nhde ined by EOC =logehl+1−logehl log(1/pNhl+1)−log(1/pNhl),l=1,2,..., (26) Mo eo e , we e alua e he “e ec i i y index” ie := η h(˜uh) ku−˜uhkX2+Nh(˜uh)21/2= ρ h(˜uh)2+Nh(˜uh)21/2 ku−˜uhkX2+Nh(˜uh)21/2.(27) Le us no e ha he index ie is no a s anda d e ec i i y index since ρ h(˜uh)is he app oxima ion o Rh(˜uh)and no o ku−˜uhkX. Ob iously, i Nh(˜uh)domina e ρ h(˜uh)and ku−˜uhkX hen he index ie is close o one. In his case i would be also in e es ing o e alua e he a io ρ h(˜uh)/ku−˜uhkX. 4.1 (E1): Nonlinea Con ec ion-Di usion Equa ion wi h a Co ne Singula i y We conside he scala nonlinea con ec ion-di usion equa ion −∇·(K(u)∇u)− ∂ u2 ∂ x1− ∂ u2 ∂ x2=gin Ω:= (0,1)2,(28) 61 .00 .25 .50 .75 1.00 .00 .25 .50 .75 1.00 .00 .01 .03 .04 .05 .00 .01 .03 .04 .05 P 0 P 0 P 1 P 1 P 2 P 2 P 3 P 3 P 4 P 4 P 5 P 5 hp Sou ce: Own Fig. 1.Example (E1) gi en by (28) –(30) wi h α =−3/2: he inal g id wi h he co - esponding deg ees o polynomial app oxima ion, he whole domain (le ) and i s de ail (0,1/20)×(0,1/20)( igh ) whe e K(u)is he nonsymme ic ma ix gi en by K(u) = ε 2+a c an(u) (2−a c an(u))/4 0(4+a c an(u))/2.(29) The pa ame e ε >0 plays a ole o an amoun o di usi i y and we pu ε =10−3. We p esc ibe a Di ichle bounda y condi ion on ∂ Ωand se he sou ce e m gsuch ha he exac solu ion is u(x1,x2) = (x2 1+x2 2) α /2x1x2(1−x1)(1−x2), α ∈R.(30) We p esen wo choices: α =4 and α =−3/2. I is possible o show (see [1]) ha u∈ H κ (Ω), κ ∈(0,3+ α ), whe e H κ (Ω)deno es he Sobole -Slobode skii space o unc ions wi h “non-in ege de i a i es”. Whe eas he choice α =4 gi es su icien ly egula solu ion, he choice α =−3/2 leads o he solu ion wi h a singula i y a x1=x2=0. Nume ical exam- ples p esen ed in [4], ca ied ou o a li le di e en p oblem, show ha his singula i y a oids o achie e an o de o con e gence be e han O(h3/2)in he L2-no m and O(h1/2)in he H1-semino m o any deg ee o polynomial app oxima ion. Ne e heless, he exac solu ion is egula ou side o he singula i y. Table 1 shows he esul s o p oblem (28) – (30) wi h α =4 and α =−3/2, namely he alues o he e o ku−˜uhkX, noncon o mi y Nh(˜uh), esiduum e o es ima e ρ h(˜uh)wi h he co esponding EOC, index ie and he compu a ional imes in seconds. We obse e ha he compu a ional e o ku−˜uhkXcon e ge exponen ially o α =4 and signi ican ly as e han O(h1/2) o α =−3/2. Fu he mo e, he index ie is e y close o one o inc easing Nhwhich suppo s he accu acy o he me hod. A small inc ease o ie in Table 1 o α =4 o he las adap a ion le el is caused by he ac ha we a e close o he machine accu acy. Fu he mo e, Figu e 1 shows he inal hp-g id ob ained wi h he aid o he hp-DGFE algo- i hm o α =−3/2. (The case α =4 is no in e es ing since only p- e inemen is ca ied ou due o he egula i y o he exac solu ion.) We obse e ha he h-adap a ion was ca ied ou in a small egion nea he singula i y. On he o he hand, he p-adap a ion appea s in egions whe e he solu ion is egula . 62 Tab. 1.Example (E1) gi en by (28) –(30): e o ku−˜uhkX, noncon o mi y Nh(˜uh), esiduum e o es ima e ρ h(˜uh)wi h he co esponding EOC, index ie and he compu a ional ime in seconds α =4: le #ThNhku−˜uhkXEOC Nh(˜uh)EOC ρ h(˜uh)EOC ie CPU(s) 0 128 384 6.45E-03 – 1.99E-02 – 5.58E-03 – 0.99 0.4 1 128 705 4.56E-04 8.72 3.07E-03 6.15 6.60E-04 7.03 1.01 0.7 2 128 384 6.45E-03 8.72 1.99E-02 6.15 5.58E-03 7.03 0.99 0.4 3 128 768 4.43E-04 7.73 3.01E-03 5.45 6.35E-04 6.27 1.01 0.7 4 128 1280 3.87E-05 9.54 2.75E-04 9.36 5.22E-05 9.78 1.01 1.0 5 128 1920 2.75E-06 13.05 1.73E-05 13.65 3.05E-06 14.02 1.00 1.4 6 128 2688 1.10E-07 19.12 7.04E-07 19.02 1.13E-07 19.59 1.00 2.2 7 128 3584 2.49E-09 26.35 1.62E-08 26.24 2.42E-09 26.72 1.00 3.4 8 128 4608 2.98E-11 35.22 2.23E-10 34.08 3.00E-11 34.92 1.00 5.3 9 128 5760 3.17E-15 82.03 2.02E-14 83.51 1.56E-14 67.83 1.25 9.0 α =−3/2: le #ThNhku−˜uhkXEOC Nh(˜uh)EOC ρ h(˜uh)EOC ie CPU(s) 0 128 384 1.32E-02 – 1.41E-01 – 4.52E-02 – 1.05 0.5 1 128 759 5.98E-03 2.32 6.70E-02 2.18 1.26E-02 3.75 1.01 0.8 2 128 919 5.50E-03 0.87 6.36E-02 0.55 6.26E-03 7.31 1.00 1.1 3 128 969 4.30E-03 9.31 5.52E-02 5.36 4.34E-03 13.81 1.00 1.4 4 134 1089 2.98E-03 6.29 3.96E-02 5.69 3.09E-03 5.86 1.00 1.6 5 140 1191 2.10E-03 7.81 2.75E-02 8.14 2.07E-03 8.91 1.00 1.9 6 152 1371 1.49E-03 4.82 1.93E-02 4.99 1.45E-03 5.03 1.00 2.1 7 158 1476 1.07E-03 9.07 1.37E-02 9.33 1.05E-03 8.72 1.00 2.5 8 164 1578 7.71E-04 9.78 9.78E-03 10.13 7.97E-04 8.34 1.00 2.9 9 176 1758 5.65E-04 5.75 7.04E-03 6.09 6.35E-04 4.21 1.00 3.3 10 188 1938 4.26E-04 5.81 5.15E-03 6.41 5.37E-04 3.43 1.00 3.8 11 200 2118 3.35E-04 5.41 3.87E-03 6.41 4.81E-04 2.48 1.00 4.3 Sou ce: Own 4.2 (E2): Linea Con ec ion-Di usion Equa ion wi h he S ong In e io Laye and he Exponen ial Bounda y Laye We conside he example om [10], gi en by − ε △u+b1 ∂ u ∂ x1+b2 ∂ u ∂ x2=0 in Ω:= (0,1)2,(31) whe e ε =10−8is a cons an di usion coe icien and (b1,b2) = (cos(− π /3),sin(− π /3)) is he con ec ion. We p esc ibe he Di ichle bounda y condi ion on ∂ Ωby uD(x1,x2) = 0 o x1=1 o x2≤0.7, 1 o he wise.(32) The solu ion possesses an in e io laye in he di ec ion o he con ec ion s a ing in (x1,x2) = (0,0.7)and con ains wo bounda y laye s along x1=0 and x2=0. On he bounda y x1=1 and on he igh pa o he bounda y x2=0, exponen ial laye s a e de eloped. The wid h o laye s a e p opo ional o ε . Fo his case, he exac solu ion is piecewise cons an excep hin egions along he bounda y and in e io laye , howe e , i s analy ical exp ession is unknown. The e o e he compu a ional 63 Tab. 2.Example (E2) gi en by (31) –(32) wi h ε =10−8: he app oxima ion o he e - o k¯u−˜uhkX, noncon o mi y Nh(˜uh), esiduum e o es ima e ρ h(˜uh)wi h he co esponding EOC, index ie and he compu a ional ime in seconds le #ThNhk¯u−˜uhkXEOC Nh(˜uh)EOC ρ h(˜uh)EOC ie CPU(s) 0 128 384 1.25E-01 – 3.58E+00 – 1.03E+00 – 1.04 0.3 1 128 635 9.56E-02 1.08 3.57E+00 0.01 7.60E-01 1.20 1.02 0.6 2 167 1211 8.06E-02 0.53 5.03E+00 -1.06 7.60E-01 -0.00 1.01 1.0 3 245 1984 7.50E-02 0.30 7.09E+00 -1.39 5.85E-01 1.06 1.00 2.2 4 467 3921 6.35E-02 0.49 1.00E+01 -1.02 5.95E-01 -0.05 1.00 4.0 5 1025 9751 6.12E-02 0.08 1.42E+01 -0.76 6.89E-01 -0.32 1.00 9.4 6 2189 21663 4.51E-02 0.76 2.01E+01 -0.87 7.32E-01 -0.15 1.00 18.2 7 4910 54765 3.17E-02 0.76 2.84E+01 -0.75 6.75E-01 0.17 1.00 50.8 8 7733 106285 2.28E-02 0.99 2.86E+01 -0.02 5.38E-01 0.68 1.00 196.6 9 14240 221582 1.76E-02 0.71 2.86E+01 -0.00 5.20E-01 0.09 1.00 505.8 10 19103 310336 1.62E-02 0.50 2.86E+01 -0.00 5.16E-01 0.04 1.00 1046.7 11 17849 272175 1.61E-02 -0.05 2.86E+01 0.00 5.16E-01 -0.00 1.00 1163.8 12 17426 250641 1.61E-02 -0.01 2.86E+01 -0.00 5.16E-01 0.00 1.00 1226.7 13 17366 240586 1.61E-02 0.03 2.86E+01 -0.01 5.16E-01 0.00 1.00 1248.5 14 17288 236116 1.61E-02 -0.16 2.86E+01 0.02 5.16E-01 0.01 1.00 1290.4 15 17207 233366 1.61E-02 0.12 2.86E+01 -0.05 5.16E-01 0.01 1.00 1311.4 Sou ce: Own e o ku−˜uhkXis app oxima ed by k¯u−˜uhkXwhe e ¯uis piecewise cons an unc ion co e- sponding o he exac solu ion o (31) in he limi wi h ε →0. (The bounda y condi ions ha e o be modi ied o cou se.) Table 2 shows he esul s o compu a ions (E2) o p oblem (31) – (32), namely he alues o he app oxima ion o he e o k¯u−˜uhkX, noncon o mi y Nh(˜uh), esiduum e o es ima e ρ h(˜uh)wi h he co esponding EOC, index ie and he compu a ional imes in seconds. The alue k¯u−˜uhkXdoes no end o ze o, p obably due o he di e ence be ween he exac solu ion uand i s app oxima ion ¯u. Mo eo e , he noncon o mi y Nh(˜uh)is high, i will dec ease wi h an addi ional mesh adap a ion, howe e , he e we ace a p oblem wi h a oo high numbe o deg ee o eedom and he limi s o ou compu e . I would be mo e e icien o apply an aniso opic mesh adap a ion. Ne e heless, he esul s show ha he p esen edhp-me hodwo ks easonably, bo h laye s a e well cap u ed. Fu he mo e, Figu e 2 shows he inal hp-g id ob ained wi h he aid o he hp-DGFE al- go i hm a e 5, 10 and 15 adap i e cycles. We obse e ha he h-adap a ion was ca ied ou in egions nea bo h laye s. In egions, whe e he solu ion is cons an , he P0app oxima ion is inally used. We also obse e ha he algo i hm is able o dec ease Nhwhen he hin laye s a e well localized. Finally, Figu e 3 shows he de ail o he diagonal cu [0,0]→[1,1]o he app oxima e solu ion a e l=5, l=10 and l=15 adap a ion cycles. Conclusion We p esen ed a hp-adap i e nume ical me hod o he solu ion o he second o de bounda y alue p oblem. This app oach is based on a heu is ic app oxima ion o he e o measu ed in a dual no m. Al hough a heo e ical jus i ica ion o his echnique is missing, nume ical expe imen s show easonable compu a ional p ope ies. 64